In the same year Lagrange gave a positive answer to the case \(k=4\)  of Warring's problem. The first proof which concerns every \(k\in {\mathbb {N}}\)  was given by Hilbert [1]} in 1909. Hilbert's proof relied on, what is today called, Hilbert's identity which follows from some algebraic properties of polynomials. In the 1920' Hardy and Ramanujan [2]} and Hardy and Littlewood [3]} began the study of questions related to Warring's problem from a more quantitative perspective. Namely, for \(N\in {\mathbb {N}}\)  by \(r_k(N)\)  we denote the number of \(d\) -tuples \((n_1, n_2, \ldots , n_d)\in {\mathbb {N}}^d\)  which solve the equation (REF ). In [2]}, [3]} the authors were interested in the asymptotic size of \(r_k(N)\)  as \(N\)  approaches infinity.
